{"paper":{"title":"Monochromatic $k$ in a row","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kuo-Han Ku","submitted_at":"2026-06-11T04:11:49Z","abstract_excerpt":"We study a variant of the $k$-in-a-row game in which players alternatively claim positions until a $k$-in-a-row is created among all claimed positions. This leads to the constraint near $k$-in-a-row avoiding on configurations and the associated problem of determining their extremal densities of such configurations.\n  We investigate this problem on two types of boards: the grid $\\mathbb{Z}^2$ and hypercubes $[k]^d$. For the grid $\\mathbb{Z}^2$, we establish nearly tight bounds on the maximum density $D(k,\\mathbb{Z}^2)$, showing that $D(k,\\mathbb{Z}^2)=1-\\frac{2}{k}$ whenever $3\\nmid k$, and det"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12880","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.12880/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}